Question: Evaluate $~~\int xe^{5x}dx\,$. Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac4{25}e^{5x}+C$ (Choice B) B $\dfrac{4x}{25}e^{5x}+C$ (Choice C) C $\dfrac x5e^{5x}-\dfrac1{25}e^{5x}+C$ (Choice D) D $\dfrac x5e^{5x}-\dfrac1{5}e^{5x}+C$
Answer: We will solve this by integrating by parts. We know that $ \int u(x)v\,^\prime(x)dx = u(x)v(x)-\int u\,^\prime(x)v(x)dx\,$. We can rewrite this as $ \int u\ dv = uv-\int v\ du\,$. In this problem we will let $~u = x~$ and $~dv=e^{5x} dx\,$. Then $~du = dx~$ and $~v = \int e^{5x}dx = \dfrac15e^{5x}$. Integration by parts gives $ \int xe^{5x}dx = x\cdot\dfrac15e^{5x}-\int\dfrac15e^{5x}dx$ $ \,=\dfrac x5e^{5x}-\dfrac1{25}e^{5x}+C\,$.